Efficient polynomial-time outer bounds on state trajectories for uncertain polynomial systems using skewed structured singular values

Outer bounds for the evolution of state trajectories of uncertain systems are useful for many purposes such as robust control design, state and parameter estimation, model invalidation, safety evaluation, fault detection and diagnosis, and experimental design. Obtaining tight outer bounds for trajectory bundles of uncertain nonlinear systems subject to disturbances is a challenging task. This paper proposes a new approach to obtaining such bounds with reasonable computational requirements for discrete-time polynomial systems with uncertain initial state and fixed and/or time-varying uncertain parameters and disturbances. The map describing the dynamics of the uncertain system is reformulated as a linear fractional transformation, which is used to bound the state trajectories by employing polynomial-time algorithms for computing upper and lower bounds for the skewed structured singular value. Three different algorithms to efficiently calculate outer bounds on the evolution of the system trajectories are presented, which have different tradeoffs between computational complexity and conservatism. The tradeoffs are illustrated in numerical examples, which demonstrate the small conservatism of the tightest bounds.